The sonic line as a free boundary

Abstract

We consider the steady transonic small disturbance equations on a domain and with data that lead to a solution that depends on a single variable. After writing down the solution, we show that it can also be found by using a hodograph transformation followed by a partial Fourier transform. This motivates considering perturbed problems that can be solved with the same technique. We identify a class of such problems. Introduction.There are many contexts in which free boundaries arise in systems of conservation laws that change type when they are formulated as steady or quasi-steady problems. The free boundary may take the form of a transonic shock. In that case, one is often presented with a situation where the flow is completely known (typically it is a constant state) on one side of the free boundary. On the free boundary itself, a set of boundary conditions is given by the Rankine-Hugoniot conditions. This could be

Singular Solutions to a (3 + 1)-D Protter-Morawetz Problem for Keldysh-Type Equations

“…Different aspects of Protter problems and several their generalizations (including some applications in the industrial explosion process) are studied by many authors (see Aldashev and Kim [18], Choi and Park [19], Aldashev [20], and references therein). For different statements of other related problems for mixed-type equations of the first kind, including nonlinear equations, see [21][22][23][24][25][26][27].…”

Singular solutions to Protter problem for Keldysh type equations

“…Find a solution U of the problem (14), (15) in D ε . (14). However, in Tricomi case β ∈ (0, 1 2 ), but right-hand side function has the form F = (η − ξ ) −4β f .…”

“…When equations are considered only in the hyperbolic part of the original Protter domain we arrived to the Protter problems in domain Ω m . For results concerning uniqueness and existence or nonexistence of nontrivial solutions to related problems for hyperbolic-elliptic type equations see [2,3,6,[14][15][16][17][18]27].…”

On uniqueness of quasi-regular solutions to Protter problem for Keldish type equations

“…Results for uniqueness are obtained by A. Aziz and M. Schneider [2], but up to now not a single example of nontrivial solution of the new problem, neither a general existence result is known. Some different statements of Darboux type problems in R 3 or some connected with them Protter problem for mixed type equations are investigated by A.Bitsadze [4], J. Barros-Neto and I. Gelfand [3], D. Lupo, C. Morawetz and K. Payne [11], D. Lupo, K. Payne, N. Popivanov [12], T. E. Moiseev [13], J. Rasiass [18], D. Edmunds, N. Popivanov [5], B. Keyfitz, A. Tesdall, K. Payne, N. Popivanov [19].…”